Doing Statistics Under Fat Tails

Research
Project started in 2015
by Nassim Nicholas Taleb and colleagues

(so
far Pasquale Cirillo,
Raphael Douady and other members of the Real World Risk Institute)

Background:
The
technical papers below are part of a systematic approach to uncover
mismeasurement of statistical metrics under fattailedness and
propose corrections and alternative tools. Conventional statistics fail
to cover fat tails; physicists who use power laws do not usually
produce statistical estimators, leading to a large —and consequential
— gap. It is not just changing the color of the dress (see discussion below). The initial aim was
to establish a network of Bourbaki-style collaborators in a
synchronized way working on the gap and injecting rigor in
policy-making and decision-making under fat tails.

Taleb, N.N., "The law of large numbers under fat
tails"(in
progress). This
the central idea; it shows where statistical inference is BS and explores more rigorous estimation of the mean of the
sum of fat-tailed random variables. A YouTube presentation here at MIT
Big Data Luncheon.

Taleb, N.N., "The mathematical foundations of the precautionary principle"(in
progress). Actually shows how the entire structure of probability in the social sciences is messed-up.

The
inequality papers (apply to all measures of concentration,
not just
inequality):

The
next two papers apply the
idea showing the flaw in using "averages" and "sums" as estimators of
inequality under fat tails, instead of maximum likelihood methods
applied to the tail exponent. Measurements of changes in inequality we
have recently witnessed (triggering active discussions) are based on
unrigorous methods; seen from these metrics, changes in inequality can
be either overly underestimated or severely exaggerated (particularly
those concerning wealth which has much fatter tails than income).

Taleb, N.N., "How to
(not) estimate Gini indices for fat tailed variables". The paper shows a severe but more
tractable problem with the Gini and proposes efficient unbiased
estimators, deriving their properties. Some have argued that "it is
only a problems for fat tails" except that Gini is a measure for fat
tailedness.

Milanovic,
B. and Taleb, N.N. Why the super-rich care more about inequality
than growth. In Progress. Policymaking errors in not realizing that
demand for assets arises from inequality, etc.

The dual distribution papers: techniques that help finding the "true (or shadow) mean" as opposed to the sample mean.

Cirillo, P. and Taleb, N.N.,
2016, "On the statistical
properties and tail risk of violent conflicts" (Physica A). Yes, the thesis by the science writer S.
Pinker on the "drop in violence" has no statistical basis. Under fat
tails, sample means are unstable and underestimate true means. The
paper proposes a method to use dual distributions, removing compact
support to apply Extreme Value Theory, and transfer parameters to the primal. Also a
novel robust approach to unreliable estimators.

Formalization of the barbell strategy using information theory
We are clueless about downside probability, particularly under fat
tails. We look at constructions with severe tail constraints and
compatible with gambler's ruin (a generalization of Kelly's criterion).

Undecidability: amply covered in Silent Risk (it is its theme), here is the formalization.

Douady, R. and Taleb, N.N. Statistical Undecidability Under
what conditions on the metadistribution of the probability measure is a
statistical formally decidable.

Power laws and stochastic tail Exponents mixtures of power laws.

TBA

Some BS
detecting papers that precede the project

Taleb, N.N.,
2014,
(On The conflation of long
volatility and fat tails),
Quantitative Finance. A strange overactive smear-campaigner,
Eric Falkenstein,
extremely innocent of probability, kept spreading all manner of
disinformation about my work. While it may have been ineffective in
stopping the spread of my ideas, the strawmanship resulted in people mistaking the tails with the scale of the distribution. This is meant to correct.

It is not changing the color of the dress

Many people know (well, sort of) what fat tails means, but in a vague
sense, believing that it is just another class of distributions than
the normal and they can think of them as, simply, other distributions
doing the same thing. Unfortunately things work differently:
The very definition of inference and confidence interval goes out of
the window. More rigor is required. To work with fat tails one has to
approach things differently, at a conceptual level. In fact one of us
found contradictions in discussions: once it is stated that a
distributions is fat-tailed, then many statements taken for granted are
no longer valid.

• The mean of the distribution will not
correspond to the sample mean. In fact there is no fat-tailed
distribution in which the mean can be properly estimated from the
sample mean.• Sharpe ratio, variance, beta and other
common finance metrics are uninformative. Variance and standard
deviations are not useable.• Correlations (in the Pearson sense)
usually do not exist, and when they do, provide little information (but
there are other forms of dependence).• Robust statistics is not robust at all.• Maximum likelihood methods work for
parameters (good news).• The gap between disconfirmatory and
confirmatory empiricism is wider than common statistics.• Principal components analysis is likely
to produce false factors.• Methods of moments fail to work.• There is no such thing as "typical"
large deviation: conditional on having a large move, such move isn’t "typical"